Fourier Heat Diffusion

∂u/∂t = α ∂²u/∂x² — eigenmode decomposition, high-k modes decay fastest

Initial Condition
The heat equation solution is u(x,t) = Σ bₖ sin(kπx/L) e^{-α(kπ/L)²t}. Each eigenmode decays as e^{-αk²t}: high wavenumber k modes disappear first, leaving only the smooth low-k components. The mode panel shows amplitude of each Fourier sine coefficient in real time.